Properties

Label 1728.2.c.c.1727.1
Level 1728
Weight 2
Character 1728.1727
Analytic conductor 13.798
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{-5})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1727.1
Root \(0.866025 + 1.11803i\) of \(x^{4} + x^{2} + 4\)
Character \(\chi\) \(=\) 1728.1727
Dual form 1728.2.c.c.1727.4

$q$-expansion

\(f(q)\) \(=\) \(q-2.23607i q^{5} -3.87298i q^{7} +O(q^{10})\) \(q-2.23607i q^{5} -3.87298i q^{7} -1.73205 q^{11} -2.00000 q^{13} -4.47214i q^{17} -6.92820 q^{23} +4.47214i q^{29} +3.87298i q^{31} -8.66025 q^{35} +4.00000 q^{37} +8.94427i q^{41} -7.74597i q^{43} -3.46410 q^{47} -8.00000 q^{49} -2.23607i q^{53} +3.87298i q^{55} -3.46410 q^{59} +4.00000 q^{61} +4.47214i q^{65} +7.74597i q^{67} -10.3923 q^{71} +5.00000 q^{73} +6.70820i q^{77} +7.74597i q^{79} -12.1244 q^{83} -10.0000 q^{85} -4.47214i q^{89} +7.74597i q^{91} +11.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 8q^{13} + 16q^{37} - 32q^{49} + 16q^{61} + 20q^{73} - 40q^{85} + 44q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) − 2.23607i − 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(6\) 0 0
\(7\) − 3.87298i − 1.46385i −0.681385 0.731925i \(-0.738622\pi\)
0.681385 0.731925i \(-0.261378\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.73205 −0.522233 −0.261116 0.965307i \(-0.584091\pi\)
−0.261116 + 0.965307i \(0.584091\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 4.47214i − 1.08465i −0.840168 0.542326i \(-0.817544\pi\)
0.840168 0.542326i \(-0.182456\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.92820 −1.44463 −0.722315 0.691564i \(-0.756922\pi\)
−0.722315 + 0.691564i \(0.756922\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.47214i 0.830455i 0.909718 + 0.415227i \(0.136298\pi\)
−0.909718 + 0.415227i \(0.863702\pi\)
\(30\) 0 0
\(31\) 3.87298i 0.695608i 0.937567 + 0.347804i \(0.113073\pi\)
−0.937567 + 0.347804i \(0.886927\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −8.66025 −1.46385
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.94427i 1.39686i 0.715678 + 0.698430i \(0.246118\pi\)
−0.715678 + 0.698430i \(0.753882\pi\)
\(42\) 0 0
\(43\) − 7.74597i − 1.18125i −0.806947 0.590624i \(-0.798881\pi\)
0.806947 0.590624i \(-0.201119\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.46410 −0.505291 −0.252646 0.967559i \(-0.581301\pi\)
−0.252646 + 0.967559i \(0.581301\pi\)
\(48\) 0 0
\(49\) −8.00000 −1.14286
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 2.23607i − 0.307148i −0.988137 0.153574i \(-0.950922\pi\)
0.988137 0.153574i \(-0.0490783\pi\)
\(54\) 0 0
\(55\) 3.87298i 0.522233i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.46410 −0.450988 −0.225494 0.974245i \(-0.572400\pi\)
−0.225494 + 0.974245i \(0.572400\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.47214i 0.554700i
\(66\) 0 0
\(67\) 7.74597i 0.946320i 0.880976 + 0.473160i \(0.156887\pi\)
−0.880976 + 0.473160i \(0.843113\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −10.3923 −1.23334 −0.616670 0.787222i \(-0.711519\pi\)
−0.616670 + 0.787222i \(0.711519\pi\)
\(72\) 0 0
\(73\) 5.00000 0.585206 0.292603 0.956234i \(-0.405479\pi\)
0.292603 + 0.956234i \(0.405479\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.70820i 0.764471i
\(78\) 0 0
\(79\) 7.74597i 0.871489i 0.900070 + 0.435745i \(0.143515\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −12.1244 −1.33082 −0.665410 0.746478i \(-0.731743\pi\)
−0.665410 + 0.746478i \(0.731743\pi\)
\(84\) 0 0
\(85\) −10.0000 −1.08465
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 4.47214i − 0.474045i −0.971504 0.237023i \(-0.923828\pi\)
0.971504 0.237023i \(-0.0761716\pi\)
\(90\) 0 0
\(91\) 7.74597i 0.811998i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 11.0000 1.11688 0.558440 0.829545i \(-0.311400\pi\)
0.558440 + 0.829545i \(0.311400\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 2.23607i − 0.222497i −0.993793 0.111249i \(-0.964515\pi\)
0.993793 0.111249i \(-0.0354850\pi\)
\(102\) 0 0
\(103\) − 7.74597i − 0.763233i −0.924321 0.381616i \(-0.875368\pi\)
0.924321 0.381616i \(-0.124632\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.19615 0.502331 0.251166 0.967944i \(-0.419186\pi\)
0.251166 + 0.967944i \(0.419186\pi\)
\(108\) 0 0
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 17.8885i − 1.68281i −0.540403 0.841406i \(-0.681728\pi\)
0.540403 0.841406i \(-0.318272\pi\)
\(114\) 0 0
\(115\) 15.4919i 1.44463i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −17.3205 −1.58777
\(120\) 0 0
\(121\) −8.00000 −0.727273
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 11.1803i − 1.00000i
\(126\) 0 0
\(127\) − 11.6190i − 1.03102i −0.856885 0.515508i \(-0.827603\pi\)
0.856885 0.515508i \(-0.172397\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −19.0526 −1.66463 −0.832315 0.554303i \(-0.812985\pi\)
−0.832315 + 0.554303i \(0.812985\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 4.47214i − 0.382080i −0.981582 0.191040i \(-0.938814\pi\)
0.981582 0.191040i \(-0.0611861\pi\)
\(138\) 0 0
\(139\) − 15.4919i − 1.31401i −0.753887 0.657004i \(-0.771823\pi\)
0.753887 0.657004i \(-0.228177\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3.46410 0.289683
\(144\) 0 0
\(145\) 10.0000 0.830455
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 2.23607i − 0.183186i −0.995797 0.0915929i \(-0.970804\pi\)
0.995797 0.0915929i \(-0.0291958\pi\)
\(150\) 0 0
\(151\) − 3.87298i − 0.315179i −0.987505 0.157589i \(-0.949628\pi\)
0.987505 0.157589i \(-0.0503723\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.66025 0.695608
\(156\) 0 0
\(157\) −20.0000 −1.59617 −0.798087 0.602542i \(-0.794154\pi\)
−0.798087 + 0.602542i \(0.794154\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 26.8328i 2.11472i
\(162\) 0 0
\(163\) − 23.2379i − 1.82013i −0.414462 0.910066i \(-0.636030\pi\)
0.414462 0.910066i \(-0.363970\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.46410 0.268060 0.134030 0.990977i \(-0.457208\pi\)
0.134030 + 0.990977i \(0.457208\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 15.6525i − 1.19004i −0.803712 0.595018i \(-0.797145\pi\)
0.803712 0.595018i \(-0.202855\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −5.19615 −0.388379 −0.194189 0.980964i \(-0.562208\pi\)
−0.194189 + 0.980964i \(0.562208\pi\)
\(180\) 0 0
\(181\) 16.0000 1.18927 0.594635 0.803996i \(-0.297296\pi\)
0.594635 + 0.803996i \(0.297296\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 8.94427i − 0.657596i
\(186\) 0 0
\(187\) 7.74597i 0.566441i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 17.3205 1.25327 0.626634 0.779314i \(-0.284432\pi\)
0.626634 + 0.779314i \(0.284432\pi\)
\(192\) 0 0
\(193\) −25.0000 −1.79954 −0.899770 0.436365i \(-0.856266\pi\)
−0.899770 + 0.436365i \(0.856266\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 24.5967i 1.75245i 0.481906 + 0.876223i \(0.339945\pi\)
−0.481906 + 0.876223i \(0.660055\pi\)
\(198\) 0 0
\(199\) 11.6190i 0.823646i 0.911264 + 0.411823i \(0.135108\pi\)
−0.911264 + 0.411823i \(0.864892\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 17.3205 1.21566
\(204\) 0 0
\(205\) 20.0000 1.39686
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) − 15.4919i − 1.06651i −0.845955 0.533254i \(-0.820969\pi\)
0.845955 0.533254i \(-0.179031\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −17.3205 −1.18125
\(216\) 0 0
\(217\) 15.0000 1.01827
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 8.94427i 0.601657i
\(222\) 0 0
\(223\) 7.74597i 0.518708i 0.965782 + 0.259354i \(0.0835097\pi\)
−0.965782 + 0.259354i \(0.916490\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.46410 0.229920 0.114960 0.993370i \(-0.463326\pi\)
0.114960 + 0.993370i \(0.463326\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 4.47214i − 0.292979i −0.989212 0.146490i \(-0.953202\pi\)
0.989212 0.146490i \(-0.0467975\pi\)
\(234\) 0 0
\(235\) 7.74597i 0.505291i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 13.8564 0.896296 0.448148 0.893959i \(-0.352084\pi\)
0.448148 + 0.893959i \(0.352084\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 17.8885i 1.14286i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 10.3923 0.655956 0.327978 0.944685i \(-0.393633\pi\)
0.327978 + 0.944685i \(0.393633\pi\)
\(252\) 0 0
\(253\) 12.0000 0.754434
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 22.3607i 1.39482i 0.716672 + 0.697410i \(0.245665\pi\)
−0.716672 + 0.697410i \(0.754335\pi\)
\(258\) 0 0
\(259\) − 15.4919i − 0.962622i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.92820 0.427211 0.213606 0.976920i \(-0.431479\pi\)
0.213606 + 0.976920i \(0.431479\pi\)
\(264\) 0 0
\(265\) −5.00000 −0.307148
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4.47214i 0.272671i 0.990663 + 0.136335i \(0.0435325\pi\)
−0.990663 + 0.136335i \(0.956467\pi\)
\(270\) 0 0
\(271\) − 11.6190i − 0.705801i −0.935661 0.352900i \(-0.885195\pi\)
0.935661 0.352900i \(-0.114805\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −20.0000 −1.20168 −0.600842 0.799368i \(-0.705168\pi\)
−0.600842 + 0.799368i \(0.705168\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 4.47214i − 0.266785i −0.991063 0.133393i \(-0.957413\pi\)
0.991063 0.133393i \(-0.0425871\pi\)
\(282\) 0 0
\(283\) 30.9839i 1.84180i 0.389799 + 0.920900i \(0.372544\pi\)
−0.389799 + 0.920900i \(0.627456\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 34.6410 2.04479
\(288\) 0 0
\(289\) −3.00000 −0.176471
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 22.3607i − 1.30632i −0.757218 0.653162i \(-0.773442\pi\)
0.757218 0.653162i \(-0.226558\pi\)
\(294\) 0 0
\(295\) 7.74597i 0.450988i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 13.8564 0.801337
\(300\) 0 0
\(301\) −30.0000 −1.72917
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 8.94427i − 0.512148i
\(306\) 0 0
\(307\) 23.2379i 1.32626i 0.748506 + 0.663129i \(0.230772\pi\)
−0.748506 + 0.663129i \(0.769228\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −27.7128 −1.57145 −0.785725 0.618576i \(-0.787710\pi\)
−0.785725 + 0.618576i \(0.787710\pi\)
\(312\) 0 0
\(313\) 5.00000 0.282617 0.141308 0.989966i \(-0.454869\pi\)
0.141308 + 0.989966i \(0.454869\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 15.6525i − 0.879131i −0.898211 0.439565i \(-0.855133\pi\)
0.898211 0.439565i \(-0.144867\pi\)
\(318\) 0 0
\(319\) − 7.74597i − 0.433691i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 13.4164i 0.739671i
\(330\) 0 0
\(331\) − 7.74597i − 0.425757i −0.977079 0.212878i \(-0.931716\pi\)
0.977079 0.212878i \(-0.0682838\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 17.3205 0.946320
\(336\) 0 0
\(337\) −10.0000 −0.544735 −0.272367 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 6.70820i − 0.363270i
\(342\) 0 0
\(343\) 3.87298i 0.209121i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 32.9090 1.76665 0.883323 0.468765i \(-0.155301\pi\)
0.883323 + 0.468765i \(0.155301\pi\)
\(348\) 0 0
\(349\) 16.0000 0.856460 0.428230 0.903670i \(-0.359137\pi\)
0.428230 + 0.903670i \(0.359137\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 17.8885i − 0.952111i −0.879415 0.476056i \(-0.842066\pi\)
0.879415 0.476056i \(-0.157934\pi\)
\(354\) 0 0
\(355\) 23.2379i 1.23334i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 31.1769 1.64545 0.822727 0.568436i \(-0.192451\pi\)
0.822727 + 0.568436i \(0.192451\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 11.1803i − 0.585206i
\(366\) 0 0
\(367\) − 27.1109i − 1.41518i −0.706625 0.707588i \(-0.749783\pi\)
0.706625 0.707588i \(-0.250217\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −8.66025 −0.449618
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 8.94427i − 0.460653i
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 24.2487 1.23905 0.619526 0.784976i \(-0.287325\pi\)
0.619526 + 0.784976i \(0.287325\pi\)
\(384\) 0 0
\(385\) 15.0000 0.764471
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 2.23607i − 0.113373i −0.998392 0.0566866i \(-0.981946\pi\)
0.998392 0.0566866i \(-0.0180536\pi\)
\(390\) 0 0
\(391\) 30.9839i 1.56692i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 17.3205 0.871489
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 17.8885i − 0.893311i −0.894706 0.446656i \(-0.852615\pi\)
0.894706 0.446656i \(-0.147385\pi\)
\(402\) 0 0
\(403\) − 7.74597i − 0.385854i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −6.92820 −0.343418
\(408\) 0 0
\(409\) −7.00000 −0.346128 −0.173064 0.984911i \(-0.555367\pi\)
−0.173064 + 0.984911i \(0.555367\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 13.4164i 0.660178i
\(414\) 0 0
\(415\) 27.1109i 1.33082i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 17.3205 0.846162 0.423081 0.906092i \(-0.360949\pi\)
0.423081 + 0.906092i \(0.360949\pi\)
\(420\) 0 0
\(421\) 4.00000 0.194948 0.0974740 0.995238i \(-0.468924\pi\)
0.0974740 + 0.995238i \(0.468924\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 15.4919i − 0.749707i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −10.3923 −0.500580 −0.250290 0.968171i \(-0.580526\pi\)
−0.250290 + 0.968171i \(0.580526\pi\)
\(432\) 0 0
\(433\) −13.0000 −0.624740 −0.312370 0.949960i \(-0.601123\pi\)
−0.312370 + 0.949960i \(0.601123\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) − 3.87298i − 0.184847i −0.995720 0.0924237i \(-0.970539\pi\)
0.995720 0.0924237i \(-0.0294614\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 24.2487 1.15209 0.576046 0.817418i \(-0.304595\pi\)
0.576046 + 0.817418i \(0.304595\pi\)
\(444\) 0 0
\(445\) −10.0000 −0.474045
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 17.8885i − 0.844213i −0.906546 0.422106i \(-0.861291\pi\)
0.906546 0.422106i \(-0.138709\pi\)
\(450\) 0 0
\(451\) − 15.4919i − 0.729487i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 17.3205 0.811998
\(456\) 0 0
\(457\) −31.0000 −1.45012 −0.725059 0.688686i \(-0.758188\pi\)
−0.725059 + 0.688686i \(0.758188\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 38.0132i 1.77045i 0.465164 + 0.885225i \(0.345995\pi\)
−0.465164 + 0.885225i \(0.654005\pi\)
\(462\) 0 0
\(463\) 3.87298i 0.179993i 0.995942 + 0.0899964i \(0.0286856\pi\)
−0.995942 + 0.0899964i \(0.971314\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −5.19615 −0.240449 −0.120225 0.992747i \(-0.538361\pi\)
−0.120225 + 0.992747i \(0.538361\pi\)
\(468\) 0 0
\(469\) 30.0000 1.38527
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 13.4164i 0.616887i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 17.3205 0.791394 0.395697 0.918381i \(-0.370503\pi\)
0.395697 + 0.918381i \(0.370503\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 24.5967i − 1.11688i
\(486\) 0 0
\(487\) − 23.2379i − 1.05301i −0.850172 0.526505i \(-0.823502\pi\)
0.850172 0.526505i \(-0.176498\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −8.66025 −0.390832 −0.195416 0.980720i \(-0.562606\pi\)
−0.195416 + 0.980720i \(0.562606\pi\)
\(492\) 0 0
\(493\) 20.0000 0.900755
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 40.2492i 1.80542i
\(498\) 0 0
\(499\) − 38.7298i − 1.73379i −0.498495 0.866893i \(-0.666114\pi\)
0.498495 0.866893i \(-0.333886\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −20.7846 −0.926740 −0.463370 0.886165i \(-0.653360\pi\)
−0.463370 + 0.886165i \(0.653360\pi\)
\(504\) 0 0
\(505\) −5.00000 −0.222497
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 11.1803i 0.495560i 0.968816 + 0.247780i \(0.0797010\pi\)
−0.968816 + 0.247780i \(0.920299\pi\)
\(510\) 0 0
\(511\) − 19.3649i − 0.856653i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −17.3205 −0.763233
\(516\) 0 0
\(517\) 6.00000 0.263880
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 22.3607i 0.979639i 0.871824 + 0.489820i \(0.162937\pi\)
−0.871824 + 0.489820i \(0.837063\pi\)
\(522\) 0 0
\(523\) − 23.2379i − 1.01612i −0.861321 0.508061i \(-0.830362\pi\)
0.861321 0.508061i \(-0.169638\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 17.3205 0.754493
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 17.8885i − 0.774839i
\(534\) 0 0
\(535\) − 11.6190i − 0.502331i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 13.8564 0.596838
\(540\) 0 0
\(541\) 4.00000 0.171973 0.0859867 0.996296i \(-0.472596\pi\)
0.0859867 + 0.996296i \(0.472596\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 8.94427i − 0.383131i
\(546\) 0 0
\(547\) − 7.74597i − 0.331194i −0.986194 0.165597i \(-0.947045\pi\)
0.986194 0.165597i \(-0.0529550\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 30.0000 1.27573
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 15.6525i − 0.663217i −0.943417 0.331608i \(-0.892409\pi\)
0.943417 0.331608i \(-0.107591\pi\)
\(558\) 0 0
\(559\) 15.4919i 0.655239i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −19.0526 −0.802970 −0.401485 0.915866i \(-0.631506\pi\)
−0.401485 + 0.915866i \(0.631506\pi\)
\(564\) 0 0
\(565\) −40.0000 −1.68281
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 35.7771i 1.49985i 0.661521 + 0.749927i \(0.269911\pi\)
−0.661521 + 0.749927i \(0.730089\pi\)
\(570\) 0 0
\(571\) 7.74597i 0.324159i 0.986778 + 0.162079i \(0.0518200\pi\)
−0.986778 + 0.162079i \(0.948180\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 46.9574i 1.94812i
\(582\) 0 0
\(583\) 3.87298i 0.160403i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 19.0526 0.786383 0.393192 0.919457i \(-0.371371\pi\)
0.393192 + 0.919457i \(0.371371\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 4.47214i − 0.183649i −0.995775 0.0918243i \(-0.970730\pi\)
0.995775 0.0918243i \(-0.0292698\pi\)
\(594\) 0 0
\(595\) 38.7298i 1.58777i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −17.3205 −0.707697 −0.353848 0.935303i \(-0.615127\pi\)
−0.353848 + 0.935303i \(0.615127\pi\)
\(600\) 0 0
\(601\) −31.0000 −1.26452 −0.632258 0.774758i \(-0.717872\pi\)
−0.632258 + 0.774758i \(0.717872\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 17.8885i 0.727273i
\(606\) 0 0
\(607\) − 7.74597i − 0.314399i −0.987567 0.157200i \(-0.949753\pi\)
0.987567 0.157200i \(-0.0502466\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 6.92820 0.280285
\(612\) 0 0
\(613\) −38.0000 −1.53481 −0.767403 0.641165i \(-0.778451\pi\)
−0.767403 + 0.641165i \(0.778451\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 17.8885i − 0.720166i −0.932920 0.360083i \(-0.882748\pi\)
0.932920 0.360083i \(-0.117252\pi\)
\(618\) 0 0
\(619\) 38.7298i 1.55668i 0.627841 + 0.778342i \(0.283939\pi\)
−0.627841 + 0.778342i \(0.716061\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −17.3205 −0.693932
\(624\) 0 0
\(625\) −25.0000 −1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 17.8885i − 0.713263i
\(630\) 0 0
\(631\) − 34.8569i − 1.38763i −0.720154 0.693815i \(-0.755929\pi\)
0.720154 0.693815i \(-0.244071\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −25.9808 −1.03102
\(636\) 0 0
\(637\) 16.0000 0.633943
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 44.7214i − 1.76639i −0.469008 0.883194i \(-0.655389\pi\)
0.469008 0.883194i \(-0.344611\pi\)
\(642\) 0 0
\(643\) 30.9839i 1.22188i 0.791675 + 0.610942i \(0.209209\pi\)
−0.791675 + 0.610942i \(0.790791\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 20.7846 0.817127 0.408564 0.912730i \(-0.366030\pi\)
0.408564 + 0.912730i \(0.366030\pi\)
\(648\) 0 0
\(649\) 6.00000 0.235521
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 11.1803i 0.437521i 0.975779 + 0.218760i \(0.0702013\pi\)
−0.975779 + 0.218760i \(0.929799\pi\)
\(654\) 0 0
\(655\) 42.6028i 1.66463i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 8.66025 0.337356 0.168678 0.985671i \(-0.446050\pi\)
0.168678 + 0.985671i \(0.446050\pi\)
\(660\) 0 0
\(661\) 16.0000 0.622328 0.311164 0.950356i \(-0.399281\pi\)
0.311164 + 0.950356i \(0.399281\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 30.9839i − 1.19970i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −6.92820 −0.267460
\(672\) 0 0
\(673\) 23.0000 0.886585 0.443292 0.896377i \(-0.353810\pi\)
0.443292 + 0.896377i \(0.353810\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 22.3607i − 0.859391i −0.902974 0.429695i \(-0.858621\pi\)
0.902974 0.429695i \(-0.141379\pi\)
\(678\) 0 0
\(679\) − 42.6028i − 1.63495i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −10.3923 −0.397650 −0.198825 0.980035i \(-0.563713\pi\)
−0.198825 + 0.980035i \(0.563713\pi\)
\(684\) 0 0
\(685\) −10.0000 −0.382080
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 4.47214i 0.170375i
\(690\) 0 0
\(691\) 15.4919i 0.589341i 0.955599 + 0.294670i \(0.0952099\pi\)
−0.955599 + 0.294670i \(0.904790\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −34.6410 −1.31401
\(696\) 0 0
\(697\) 40.0000 1.51511
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 15.6525i − 0.591186i −0.955314 0.295593i \(-0.904483\pi\)
0.955314 0.295593i \(-0.0955172\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −8.66025 −0.325702
\(708\) 0 0
\(709\) −32.0000 −1.20179 −0.600893 0.799330i \(-0.705188\pi\)
−0.600893 + 0.799330i \(0.705188\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 26.8328i − 1.00490i
\(714\) 0 0
\(715\) − 7.74597i − 0.289683i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −20.7846 −0.775135 −0.387568 0.921841i \(-0.626685\pi\)
−0.387568 + 0.921841i \(0.626685\pi\)
\(720\) 0 0
\(721\) −30.0000 −1.11726
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 42.6028i 1.58005i 0.613074 + 0.790026i \(0.289933\pi\)
−0.613074 + 0.790026i \(0.710067\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −34.6410 −1.28124
\(732\) 0 0
\(733\) −8.00000 −0.295487 −0.147743 0.989026i \(-0.547201\pi\)
−0.147743 + 0.989026i \(0.547201\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 13.4164i − 0.494200i
\(738\) 0 0
\(739\) 23.2379i 0.854820i 0.904058 + 0.427410i \(0.140574\pi\)
−0.904058 + 0.427410i \(0.859426\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 24.2487 0.889599 0.444799 0.895630i \(-0.353275\pi\)
0.444799 + 0.895630i \(0.353275\pi\)
\(744\) 0 0
\(745\) −5.00000 −0.183186
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 20.1246i − 0.735337i
\(750\) 0 0
\(751\) 3.87298i 0.141327i 0.997500 + 0.0706636i \(0.0225117\pi\)
−0.997500 + 0.0706636i \(0.977488\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −8.66025 −0.315179
\(756\) 0 0
\(757\) 34.0000 1.23575 0.617876 0.786276i \(-0.287994\pi\)
0.617876 + 0.786276i \(0.287994\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 17.8885i − 0.648459i −0.945978 0.324230i \(-0.894895\pi\)
0.945978 0.324230i \(-0.105105\pi\)
\(762\) 0 0
\(763\) − 15.4919i − 0.560846i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6.92820 0.250163
\(768\) 0 0
\(769\) 11.0000 0.396670 0.198335 0.980134i \(-0.436447\pi\)
0.198335 + 0.980134i \(0.436447\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 4.47214i 0.160852i 0.996761 + 0.0804258i \(0.0256280\pi\)
−0.996761 + 0.0804258i \(0.974372\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 18.0000 0.644091
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 44.7214i 1.59617i
\(786\) 0 0
\(787\) − 15.4919i − 0.552228i −0.961125 0.276114i \(-0.910953\pi\)
0.961125 0.276114i \(-0.0890467\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −69.2820 −2.46339
\(792\) 0 0
\(793\) −8.00000 −0.284088
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 42.4853i − 1.50491i −0.658646 0.752453i \(-0.728870\pi\)
0.658646 0.752453i \(-0.271130\pi\)
\(798\) 0 0
\(799\) 15.4919i 0.548065i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −8.66025 −0.305614
\(804\) 0 0
\(805\) 60.0000 2.11472
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 4.47214i − 0.157232i −0.996905 0.0786160i \(-0.974950\pi\)
0.996905 0.0786160i \(-0.0250501\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −51.9615 −1.82013
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 4.47214i 0.156079i 0.996950 + 0.0780393i \(0.0248660\pi\)
−0.996950 + 0.0780393i \(0.975134\pi\)
\(822\) 0 0
\(823\) 27.1109i 0.945026i 0.881324 + 0.472513i \(0.156653\pi\)
−0.881324 + 0.472513i \(0.843347\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −31.1769 −1.08413 −0.542064 0.840337i \(-0.682357\pi\)
−0.542064 + 0.840337i \(0.682357\pi\)
\(828\) 0 0
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 35.7771i 1.23960i
\(834\) 0 0
\(835\) − 7.74597i − 0.268060i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 38.1051 1.31553 0.657767 0.753221i \(-0.271501\pi\)
0.657767 + 0.753221i \(0.271501\pi\)
\(840\) 0 0
\(841\) 9.00000 0.310345
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 20.1246i 0.692308i
\(846\) 0 0
\(847\) 30.9839i 1.06462i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −27.7128 −0.949983
\(852\) 0 0
\(853\) −2.00000 −0.0684787 −0.0342393 0.999414i \(-0.510901\pi\)
−0.0342393 + 0.999414i \(0.510901\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 4.47214i − 0.152765i −0.997079 0.0763826i \(-0.975663\pi\)
0.997079 0.0763826i \(-0.0243370\pi\)
\(858\) 0 0
\(859\) − 15.4919i − 0.528578i −0.964444 0.264289i \(-0.914863\pi\)
0.964444 0.264289i \(-0.0851373\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 20.7846 0.707516 0.353758 0.935337i \(-0.384904\pi\)
0.353758 + 0.935337i \(0.384904\pi\)
\(864\) 0 0
\(865\) −35.0000 −1.19004
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 13.4164i − 0.455120i
\(870\) 0 0
\(871\) − 15.4919i − 0.524924i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −43.3013 −1.46385
\(876\) 0 0
\(877\) 34.0000 1.14810 0.574049 0.818821i \(-0.305372\pi\)
0.574049 + 0.818821i \(0.305372\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 22.3607i 0.753350i 0.926345 + 0.376675i \(0.122933\pi\)
−0.926345 + 0.376675i \(0.877067\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −38.1051 −1.27944 −0.639722 0.768606i \(-0.720951\pi\)
−0.639722 + 0.768606i \(0.720951\pi\)
\(888\) 0 0
\(889\) −45.0000 −1.50925
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 11.6190i 0.388379i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −17.3205 −0.577671
\(900\) 0 0
\(901\) −10.0000 −0.333148
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 35.7771i − 1.18927i
\(906\) 0 0
\(907\) 15.4919i 0.514401i 0.966358 + 0.257201i \(0.0828001\pi\)
−0.966358 + 0.257201i \(0.917200\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −45.0333 −1.49202 −0.746010 0.665934i \(-0.768033\pi\)
−0.746010 + 0.665934i \(0.768033\pi\)
\(912\) 0 0
\(913\) 21.0000 0.694999
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 73.7902i 2.43677i
\(918\) 0 0
\(919\) 11.6190i 0.383274i 0.981466 + 0.191637i \(0.0613796\pi\)
−0.981466 + 0.191637i \(0.938620\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 20.7846 0.684134
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 44.7214i − 1.46726i −0.679549 0.733630i \(-0.737825\pi\)
0.679549 0.733630i \(-0.262175\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 17.3205 0.566441
\(936\) 0 0
\(937\) −31.0000 −1.01273 −0.506363 0.862320i \(-0.669010\pi\)
−0.506363 + 0.862320i \(0.669010\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 42.4853i − 1.38498i −0.721427 0.692490i \(-0.756513\pi\)
0.721427 0.692490i \(-0.243487\pi\)
\(942\) 0 0
\(943\) − 61.9677i − 2.01795i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 50.2295 1.63224 0.816119 0.577883i \(-0.196121\pi\)
0.816119 + 0.577883i \(0.196121\pi\)
\(948\) 0 0
\(949\) −10.0000 −0.324614
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 35.7771i 1.15893i 0.814996 + 0.579467i \(0.196739\pi\)
−0.814996 + 0.579467i \(0.803261\pi\)
\(954\) 0 0
\(955\) − 38.7298i − 1.25327i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −17.3205 −0.559308
\(960\) 0 0
\(961\) 16.0000 0.516129
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 55.9017i 1.79954i
\(966\) 0 0
\(967\) 27.1109i 0.871827i 0.899989 + 0.435914i \(0.143575\pi\)
−0.899989 + 0.435914i \(0.856425\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −36.3731 −1.16727 −0.583634 0.812017i \(-0.698370\pi\)
−0.583634 + 0.812017i \(0.698370\pi\)
\(972\) 0 0
\(973\) −60.0000 −1.92351
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 49.1935i 1.57384i 0.617055 + 0.786920i \(0.288325\pi\)
−0.617055 + 0.786920i \(0.711675\pi\)
\(978\) 0 0
\(979\) 7.74597i 0.247562i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −24.2487 −0.773414 −0.386707 0.922203i \(-0.626387\pi\)
−0.386707 + 0.922203i \(0.626387\pi\)
\(984\) 0 0
\(985\) 55.0000 1.75245
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 53.6656i 1.70647i
\(990\) 0 0
\(991\) − 11.6190i − 0.369088i −0.982824 0.184544i \(-0.940919\pi\)
0.982824 0.184544i \(-0.0590808\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 25.9808 0.823646
\(996\) 0 0
\(997\) −14.0000 −0.443384 −0.221692 0.975117i \(-0.571158\pi\)
−0.221692 + 0.975117i \(0.571158\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1728.2.c.c.1727.1 4
3.2 odd 2 inner 1728.2.c.c.1727.3 4
4.3 odd 2 inner 1728.2.c.c.1727.2 4
8.3 odd 2 108.2.b.a.107.2 yes 4
8.5 even 2 108.2.b.a.107.4 yes 4
12.11 even 2 inner 1728.2.c.c.1727.4 4
24.5 odd 2 108.2.b.a.107.1 4
24.11 even 2 108.2.b.a.107.3 yes 4
72.5 odd 6 324.2.h.d.107.4 8
72.11 even 6 324.2.h.d.215.1 8
72.13 even 6 324.2.h.d.107.1 8
72.29 odd 6 324.2.h.d.215.2 8
72.43 odd 6 324.2.h.d.215.4 8
72.59 even 6 324.2.h.d.107.3 8
72.61 even 6 324.2.h.d.215.3 8
72.67 odd 6 324.2.h.d.107.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.2.b.a.107.1 4 24.5 odd 2
108.2.b.a.107.2 yes 4 8.3 odd 2
108.2.b.a.107.3 yes 4 24.11 even 2
108.2.b.a.107.4 yes 4 8.5 even 2
324.2.h.d.107.1 8 72.13 even 6
324.2.h.d.107.2 8 72.67 odd 6
324.2.h.d.107.3 8 72.59 even 6
324.2.h.d.107.4 8 72.5 odd 6
324.2.h.d.215.1 8 72.11 even 6
324.2.h.d.215.2 8 72.29 odd 6
324.2.h.d.215.3 8 72.61 even 6
324.2.h.d.215.4 8 72.43 odd 6
1728.2.c.c.1727.1 4 1.1 even 1 trivial
1728.2.c.c.1727.2 4 4.3 odd 2 inner
1728.2.c.c.1727.3 4 3.2 odd 2 inner
1728.2.c.c.1727.4 4 12.11 even 2 inner